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For a plane curve define by the equation $f = f_h + f_{h+1} + \ldots + f_n$, where the $f_i$ are the homogeneous parts of degree $i$ (in the variables $x - a$ and $y - b$), and $f_h$ is the first nonzero homogeneous term, the tangent cone of the variety at $(a,b)$ is defined to be the cone $Z(f_h)$.

Suppose that we are working over $\mathbb{C}$, and that for concreteness $f = x^2 - y^2 + x^3$, $X = Z(f)$. Then the tangent cone at the origin is given by $(x - i y) (x + iy) = 0$. From the real picture, it seems like the vectors in the tangent cone are approximated by the vectors in the nearby tangent spaces.

I am wondering if it is possible to arrive at the definition of the tangent cone of $X$ at $0$ by considering something like limits of velocity vectors of suitable curves that approach the origin along $X$. Of course, these limits have no right to exist necessarily. A possibly equivalent way to ask this question is to ask if the tangent cone is union of the topological boundary (in the Grassmanian) of the tangent spaces of points (analytically) nearby.

At least for curves, I know that the tangent cone is obtained by projecting the tangent spaces of the blow-up along that point. This seems like a reasonable partial answer.

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